Definitions

Cone-shape distribution function

In mathematics, cone-shape distribution function is one of the members of Cohen's class distribution function. It was first proposed by Yunxin Zhao, Lee. E. Atlas, and Robert J. Marks in 1990. The reason why this distribution is so named is because its kernel function in $t, tau$ domain looks like two cones. The advantage of this special kernel function is that it can completely remove the cross-term between two components that have same center frequency, but on the other hand, the cross-term results form components with the same time center can not be removed by the cone-shape kernel.

Mathematical definition

The definition of the cone-shape distribution function is shown as follows:

$C_x\left(t, f\right)=int_\left\{-infty\right\}^\left\{infty\right\}int_\left\{-infty\right\}^\left\{infty\right\}A_x\left(eta,tau\right)Phi\left(eta,tau\right)exp \left(j2pi\left(eta t-tau f\right)\right), deta, dtau,$

where

$A_x\left(eta,tau\right)=int_\left\{-infty\right\}^\left\{infty\right\}x\left(t+tau /2\right)x^*\left(t-tau /2\right)e^\left\{-j2pi teta\right\}, dt,$

and the kernel function is

$Phi left\left(eta,tau right\right) = frac\left\{sin left\left(pi eta tau right\right)\right\}\left\{ pi eta tau \right\}exp left\left(-2pi alpha tau^2 right\right).$

The kernel function in $t, tau$ domain is defined as:

$phi left\left(t,tau right\right) = begin\left\{cases\right\} frac\left\{1\right\}\left\{tau\right\} exp left\left(-2pi alpha tau^2 right\right), & |tau | ge 2|t|, 0, & mbox\left\{otherwise\right\}. end\left\{cases\right\}$

Following are the magnitude distribution of the kernel function in $t, tau$ domain.

Following are the magnitude distribution of the kernel function in $eta, tau$ domain with different $alpha$ values.

As we can see from the figure above, a properly chosen kernel of cone-shape distribution function can filter out the interference on the $tau$ axis in the $eta, tau$ domain, or the ambiguity domain. Therefore, unlike the Choi-Williams distribution function, the cone-shape distribution function can effectively reduce the cross-term results form two component with same center frequency. However, the cross-terms on the $eta$ axis are still preserved.